A note on function algebras on disks

TL;DR

This paper investigates the rational convexity of certain function algebras on disks, showing that the algebra generated by specific functions equals all continuous functions on small enough disks, contrasting previous results on polynomial convexity.

Contribution

It provides conditions for rational convexity of unions of compact sets in complex space and demonstrates that the rational function algebra generated by particular functions is dense in continuous functions on small disks.

Findings

Rational convexity conditions for unions of compact sets

Equality of rational function algebra and continuous functions on small disks

Contrast with polynomial convexity results in prior work

Abstract

Let $D$ be a closed disk in the complex plane centered at the origin, $f, g$ complex valued continuous function on $D$. Let $P[f,g; D]$ (res. $R[f, g; D])$) be the uniform closure on $D$ of polynomials (res. rational functions) in variables $f$ and $g$. In \cite{OS}, using complex dynamical systems, O'Farrell and Sanabria-Garcia proved that $\{\Big(z^2, \cfrac{\overline z}{1+\overline{z}}\Big): z\in D\}$ is not polynomially convex with $D$ small enough and so that $P[z^2,\cfrac{\overline z}{1+\overline z}; D]\ne C(D)$ if $D$ is sufficient small. In this paper, we first give a certain conditions for rational convexity of union of two compact set of $\Bbb C^n$ and apply to show that $R[z^2, \cfrac{\overline z}{1+\overline z}; D]= C(D)$ for all $D$ small enough